in the fields of mechanism design and social choice theory, gibbard's theorem is a result proven by philosopher allan gibbard in 1973. it states that for any deterministic process of collective decision, at least one of the following three properties must hold:
- the process is dictatorial, i.e. there exists a distinguished agent who can impose the outcome;
- the process limits the possible outcomes to two options only;
- the process encourages agents to think strategically: once an agent has identified their preferences, they have no action at their disposal that would best defend their opinions in any situation.
a corollary of this theorem is gibbard–satterthwaite theorem about voting rules. the main difference between the two is that gibbard–satterthwaite theorem is limited to ranked (ordinal) voting rules: a voter's action consists in giving a preference ranking over the available options. gibbard's theorem is more general and considers processes of collective decision that may not be ordinal: for example, voting systems where voters assign grades to candidates.
gibbard's theorem is itself generalized by
gibbard's 1978 theorem and
hylland's theorem, which extend these results to non-deterministic processes, i.e. where the outcome may not only depend on the agents' actions but may also involve a part of chance.